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Step * 2 1 1 of Lemma free-dlwc-basis


1. T : Type
2. eq : EqDecider(T)
3. Cs : T ⟶ fset(fset(T))
4. x : Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))
5. x = \/(λs.{s}"(x)) ∈ Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))
6. deq-fset(deq-fset(eq)) ∈ EqDecider(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))
7. x ∈ fset(fset(T))
⊢ λs./\(λx.free-dlwc-inc(eq;a.Cs[a];x)"(s))"(x)
= λs.{s}"(x)
∈ fset(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))
BY
{ GenConcl ⌜x = y ∈ fset({s:fset(T)| s ∈ x} )⌝⋅ }

1
.....wf..... 
1. T : Type
2. eq : EqDecider(T)
3. Cs : T ⟶ fset(fset(T))
4. x : Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))
5. x = \/(λs.{s}"(x)) ∈ Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))
6. deq-fset(deq-fset(eq)) ∈ EqDecider(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))
7. x ∈ fset(fset(T))
⊢ x ∈ fset({s:fset(T)| s ∈ x} )

2
1. T : Type
2. eq : EqDecider(T)
3. Cs : T ⟶ fset(fset(T))
4. x : Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))
5. x = \/(λs.{s}"(x)) ∈ Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))
6. deq-fset(deq-fset(eq)) ∈ EqDecider(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))
7. x ∈ fset(fset(T))
8. y : fset({s:fset(T)| s ∈ x} )
9. x = y ∈ fset({s:fset(T)| s ∈ x} )
⊢ λs./\(λx.free-dlwc-inc(eq;a.Cs[a];x)"(s))"(y)
= λs.{s}"(y)
∈ fset(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))

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Latex:

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Latex:

1.  T  :  Type
2.  eq  :  EqDecider(T)
3.  Cs  :  T  {}\mrightarrow{}  fset(fset(T))
4.  x  :  Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))
5.  x  =  \mbackslash{}/(\mlambda{}s.\{s\}"(x))
6.  deq-fset(deq-fset(eq))  \mmember{}  EqDecider(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))
7.  x  \mmember{}  fset(fset(T))
\mvdash{}  \mlambda{}s./\mbackslash{}(\mlambda{}x.free-dlwc-inc(eq;a.Cs[a];x)"(s))"(x)  =  \mlambda{}s.\{s\}"(x)


By

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Latex:
GenConcl  \mkleeneopen{}x  =  y\mkleeneclose{}\mcdot{}

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